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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
%\title{Worst-Case Response Time Analysis of Sporadic Self-Suspending Tasks with Fixed Priorities}
\title{\LARGE{\textbf{Timing Analysis of Fixed Priority Self-Suspending Sporadic Tasks: Appendix}}}


% author names and affiliations
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%\author{\IEEEauthorblockN{Geoffrey Nelissen, Jos\'e Fonseca, Gurulingesh Raravi, and Vincent Nelis}
%\IEEEauthorblockA{CISTER/INESC-TEC Research Center\\
%ISEP, Polytechnic Institute of Porto, Portugal\\
%Email: \{grrpn, guhri, jcnfo, nelis\}@isep.ipp.pt}}
\author{\IEEEauthorblockN{Geoffrey Nelissen\IEEEauthorrefmark{1},
Jos\'e Fonseca\IEEEauthorrefmark{1}, Gurulingesh Raravi\IEEEauthorrefmark{2} and Vincent N\'elis\IEEEauthorrefmark{1}}
\IEEEauthorblockA{\IEEEauthorrefmark{1}CISTER/INESC-TEC, ISEP, Polytechnic Institute of Porto, Portugal}
\IEEEauthorblockA{\IEEEauthorrefmark{2}Xerox Research Center India, Bengaluru\\ 
Email: \IEEEauthorrefmark{1}\{grrpn, jcnfo, nelis\}@isep.ipp.pt, \IEEEauthorrefmark{2}gurulingesh.raravi@xerox.com}}

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\appendices

\section{Counter-example to the fixed-priority timing analysis for multiprocessor presented in \cite{Liu:ECRTS13}}

In this section, we present a counter-example to the fixed-priority suspension aware analysis for multiprocessor presented in \cite{Liu:ECRTS13}, thereby proving its incorrectness. 

In~\cite{Liu:ECRTS13}, assuming a multiprocessor platform composed of $m$ cores, the worst-case response time of a sporadic self-suspending task $\tau_l$ defined by its total worst-case execution time $C_l \equals \sum_{k=1}^{m_l} C_{l,k}$ and total suspension $S_l \equals \sum_{k=1}^{m_l-1} S_{l,k}$ is computed with the following theorem:

\begin{theorem}[from \cite{Liu:ECRTS13}]
Let $\psi_l$ be the set of minimum solutions of
\eqref{eq:th_liu} for $L$ below for each value of $s_{l,j} \in \{0; 1; 2;...; S_l \}$ by performing a fixed-point iteration on the RHS of \eqref{eq:th_liu} starting with $L = C_l + s_{l,j}$:
\begin{equation}
L = \left\lfloor \frac{\Omega_l(L)}{m} \right\rfloor + C_l + s_{l,j}
\label{eq:th_liu}
\end{equation}
Then $\psi_l^{\max{}} + \kappa_l$ upper-bounds $\tau_l$'s response time, where $\psi_l^{\max{}}$ is the maximum value in $\psi_l$. \qed
\label{th:liu}
\end{theorem}
$\kappa_l$ is defined in \cite{Liu:ECRTS13} and its value is different from $0$ only for soft real-time tasks.

$\Omega_l(L)$ is the total interference bound on $\tau_l$ and is given by
\begin{align}
\Omega_l(L) \equals & \sum_{\tau_i \in \tau_{hp}^s} \max\left( I_l^c(\tau_i, L),I_l^{nc}(\tau_i, L) \right) \nonumber \\
& + \sum_{\tau_i \in \tau_{hp}^e} I_l^{nc}(\tau_i, L) + \beta_{\tau_i \in \tau_{hp}^e}^{\min(m-1, n_{hp}^e)}
\label{eq:omega_big}
\end{align}
with $\tau_{hp}^s$ and $\tau_{hp}^e$ being the set of self-suspending and  non-self-suspending tasks, respectively, $n_{hp}^e$ being the number of tasks in $\tau_{hp}^e$ and $\beta_{\tau_i \in \tau_{hp}^e}^{\min(m-1, n_{hp}^e)}$ denoting the $\min(m-1, n_{hp}^e)$ greatest values of $\max(0;  I_l^{c}(\tau_i, L) - I_l^{nc}(\tau_i, L))$ for
any computational task $\tau_i \in \tau_{hp}^e$.

Note that if all tasks are non-self-suspending and $m=1$ (i.e., there is only one processor), Eq.~\eqref{eq:omega_big} becomes
\begin{equation}
\Omega_l(L) \equals \sum_{\tau_i \in \tau_{hp}^e} I_l^{nc}(\tau_i, L) 
\label{eq:omega_simp}
\end{equation}

$I_l^{c}(\tau_i, L)$ and $I_l^{nc}(\tau_i, L)$ represent the interference caused by $\tau_i$ on $\tau_l$ accounting or not for a carry-in job. They are defined as follows
\begin{small}
\begin{equation}
I_l^{nc}(\tau_i, L) \equals 
\begin{cases}
\min( \omega^{nc}(\tau_i, L), L - C_l - s_{l,j} + 1 ) &  \text{if}~ i \neq l\\
\min( \omega^{nc}(\tau_l, L) - C_l, L - C_l - s_{l,j} + 1 ) & \text{if}~ i=l \\
\end{cases}
\label{eq:Inc}
\end{equation}
\end{small}

\begin{small}
\begin{equation}
I_l^{c}(\tau_i, L) \equals 
\begin{cases}
\min( \omega^{c}(\tau_i, L), L - C_l - s_{l,j} + 1 ) &  \text{if}~ i \neq l\\
\min( \omega^{c}(\tau_l, L) - e_l, L - C_l - s_{l,j} + 1 ) & \text{if}~ i=l \\
\end{cases}
\end{equation}
\end{small}
where $\omega^{c}(\tau_i, L)$ and $\omega^{nc}(\tau_i, L)$ are the maximum workload generated by $\tau_i$ in the interval of length $L$ accounting or not for a carry-in job. They are given by the following equations:
\begin{equation}
\omega^{nc}(\tau_i, L) \equals \left( \left\lfloor \frac{L-C_i}{T_i} \right\rfloor + 1 \right) \times C_i
\label{eq:omega}
\end{equation}
\begin{equation}
\omega^{c}(\tau_i, L) \equals \Delta(\tau_i, L-C_i+D_i)
\end{equation}
where $\Delta(\tau_i, x)$ is a function defined in~\cite{Liu:ECRTS13}.

Now, we provide a counter-example to Theorem~\ref{th:liu}.

\begin{Example}
Let two tasks $\tau_l$ and $\tau_i$ execute on a single processor platform, i.e., $m=1$. Consider the case where the self-suspending region $S_l$ of $\tau_l$ is equal to $0$, its worst-case execution time is given by $C_l = 3$ and its period by $T_l = 100$. $\tau_l$ is then equivalent to a non-self-suspending task. Assume that the non-self-suspending task $\tau_i \equals \left\langle (2), 4, 4 \right\rangle$ has a higher priority than $\tau_l$. We can use Theorem~\ref{th:liu} to compute the WCRT of $\tau_l$. 

First, we observe that for $\tau_l$, as long as $L$ is smaller than $T_l + C_l = 103$, Eq.~\ref{eq:omega} is equal to $C_l$ and hence Eq.~\ref{eq:Inc} always returns $0$. 

Then, starting with $L = 3$ as prescribed by Theorem~\ref{th:liu}, we inject $L$ in Eq.~\eqref{eq:omega} for $\tau_i$, which gives
$$\omega^{nc}(\tau_i, 3) = \left( \left\lfloor \frac{3-2}{4} \right\rfloor + 1 \right) \times 2 = 2$$
Using Eq.~\ref{eq:Inc}, we get
$$I_l^{nc}(\tau_i, 3) = \min( 2, 3-3+1 ) = 1$$
Therefore, according to Eq.~\eqref{eq:omega_simp}, $\Omega(L) = 1$ and using Eq.~\eqref{eq:th_liu}, it gives
$$L = \left\lfloor \frac{1}{1} \right\rfloor + 3 + 0 =4$$

Following the same process for a second iteration, we get
$$\omega^{nc}(\tau_i, 4) = \left( \left\lfloor \frac{4-2}{4} \right\rfloor + 1 \right) \times 2 = 2$$
resulting in
$$I_l^{nc}(\tau_i, 4) = \min( 2, 4-3+1 ) = 2$$
implying $\Omega(L) = 2$ and thus
$$L = \left\lfloor \frac{2}{1} \right\rfloor + 3 + 0 =5$$

A last iteration provides
$$\omega^{nc}(\tau_i, 5) = \left( \left\lfloor \frac{5-2}{4} \right\rfloor + 1 \right) \times 2 = 2$$
$$I_l^{nc}(\tau_i, 5) = \min( 2, 5-3+1 ) = 2$$
Finally, giving $\Omega(L) = 2$ and thus
$$L = \left\lfloor \frac{2}{1} \right\rfloor + 3 + 0 =5$$

As a result, according to Theorem~\ref{th:liu}, the WCRT of $\tau_l$ is $5$.

However, it is easy to see on Fig.~\ref{fig:liu} that the WCRT of $\tau_l$ is actually $7$.
\end{Example}

\begin{figure}[t]
\centering
\includegraphics[width=0.6\columnwidth]{figures/appendix.pdf}
\caption{Counter-example to Theorem~\ref{th:liu} in~\cite{Liu:ECRTS13}}
\label{fig:liu}
\end{figure}

The example provided above proves that the analysis presented in~\cite{Liu:ECRTS13} is incorrect. Since it does not compute correctly the worst-case response time of two non-self-suspending tasks on uniprocessor, it is easy to infer its incorrectness for multiprocessor systems as well as self-suspending tasks.


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